Convergence, stability, and error analysis of the method of lines for solving loaded parabolic equations

Anar Assanova; Zhenhai Liu; Saule Kuanysh; Zhazira Kadirbayeva; Anar Assanova; Zhenhai Liu; Saule Kuanysh; Zhazira Kadirbayeva

doi:10.3934/math.20251293

AIMS Mathematics

2025, Volume 10, Issue 12: 29454-29469. doi: 10.3934/math.20251293

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Convergence, stability, and error analysis of the method of lines for solving loaded parabolic equations

1.
Institute of Mathematics and Mathematical Modeling, Department of Differential equations and Dynamical systems, 28 Shevchenko str, Almaty A26G7T5, Kazakhstan
2.
Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, Guangxi, Nanning 530006, China
3.
Al-Farabi Kazakh National University, 71 al-Farabi Ave, Almaty 050040, Kazakhstan
4.
Kazakh National Women's Teacher Training University, Department of Mathematics, 99 Ayteke bi str, Almaty A05M0T6, Kazakhstan
5.
International Information Technology University, Department of Mathematical Computer Modeling, 34/1 Manas str, Almaty A15M0F0, Kazakhstan

Received: 27 July 2025 Revised: 05 December 2025 Accepted: 08 December 2025 Published: 15 December 2025
MSC : 35K99, 65M20, 34A45, 34B10

This paper studies the method of lines for solving loaded parabolic equations and provides a theoretical analysis of its convergence, stability, and error estimates. The spatial discretization reduces the original equation to a system of loaded ordinary differential equations solved by the Dzhumabaev parameterization method. Sufficient conditions for the existence and uniqueness of the solution are established, and it is proved that the method achieves second-order accuracy in space and stable convergence. A numerical example confirms the efficiency and reliability of the proposed approach.
- loaded parabolic equations,
- nonlocal problem,
- method of lines,
- system of loaded ordinary differential equations,
- Dzhumabaev's parameterization method,
- convergence analysis
Citation: Anar Assanova, Zhenhai Liu, Saule Kuanysh, Zhazira Kadirbayeva. Convergence, stability, and error analysis of the method of lines for solving loaded parabolic equations[J]. AIMS Mathematics, 2025, 10(12): 29454-29469. doi: 10.3934/math.20251293

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Abstract

This paper studies the method of lines for solving loaded parabolic equations and provides a theoretical analysis of its convergence, stability, and error estimates. The spatial discretization reduces the original equation to a system of loaded ordinary differential equations solved by the Dzhumabaev parameterization method. Sufficient conditions for the existence and uniqueness of the solution are established, and it is proved that the method achieves second-order accuracy in space and stable convergence. A numerical example confirms the efficiency and reliability of the proposed approach.

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